Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation

Peherstorfer, Benjamin; Gómez, Pablo; Bungartz, Hans‐Joachim

doi:10.1007/s10444-015-9421-4

Cited by 9 publications

(4 citation statements)

References 47 publications

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“…The POD with Galerkin projection is the most known and frequently used reduced-order modeling method. POD based reduced-order modeling is investigated for pricing European option and American option wherein different from European option there exist variational inequalities [5,30,31] to be handled, under Black-Scholes and Heston model [2,4,20,21]. Reduced-order models based on POD are optimal in terms of energy content.…”

Section: Introductionmentioning

confidence: 99%

Reduced-Order Modeling for Heston Stochastic Volatility Model

KOZPINAR,

UZUNCA,

KARASÖZEN

2024

Hacettepe Journal of Mathematics and Statistics

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In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Heston's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, European and digital call options reveal that in general DMD requires more modes than the POD modes for the same level of accuracy. However, the speed-up factors are much higher for DMD than POD due to the non-intrusive nature of the DMD.

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Section: Introductionmentioning

confidence: 99%

Reduced-Order Modeling for Heston Stochastic Volatility Model

KOZPINAR,

UZUNCA,

KARASÖZEN

2024

Hacettepe Journal of Mathematics and Statistics

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“…Further, the resulting non-linear truncated problem has been solved by a semiimplicit FEM. In [43], Peherstorfer et al suggested reduced models for pricing basket options with the Black-Scholes and the Heston models. Shcherbakov [48] extended the operator splitting formulation to the radial basis function partition of unity method.…”

Section: Introductionmentioning

confidence: 99%

Modulus-based Successive Overrelaxation Iteration Method for Pricing American Options with the Two-asset Black–Scholes and Heston's Models Based on Finite Volume Discretization

Gan

Chen

2022

Taiwanese J. Math.

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In this paper we introduce a new numerical method for the linear complementarity problems (LCPs) arising from two-asset Black-Scholes and Heston's stochastic volatility American options pricing. Based on barycenter dual mesh, a class of finite volume method (FVM) is proposed for the spatial discretization, coupled with the backward Euler and Crank-Nicolson schemes are employed for time stepping of the partial differential equations (PDEs). Then, for the resulting time-dependent LCPs are solved by using an efficient modulus-based successive overrelaxation (MSOR) iteration method. Numerical experiments are carried out to verify the efficiency and usefulness of the proposed method.

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“…However, little attention has been paid to applications in finance. Recent work on model reduction techniques in finance with a primary focus on POD methods include, e.g., [58,56,57,49,46]. First results for RBM can be found in [16,48,49].…”

Section: Introductionmentioning

confidence: 99%

Complexity reduction for calibration to American options

Burkovska¹,

Glau²,

Mahlstedt³

et al. 2019

JCF

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American put options are among the most frequently traded single stock options, and their calibration is computationally challenging since no closed-form expression is available. Due to the higher flexibility in comparison to European options, the mathematical model involves additional constraints, and a variational inequality is obtained. We use the Heston stochastic volatility model to describe the price of a single stock option. In order to speed up the calibration process, we apply two model reduction strategies. Firstly, a reduced basis method (RBM) is used to define a suitable low-dimensional basis for the numerical approximation of the parameter-dependent partial differential equation (µPDE) model. By doing so the computational complexity for solving the µPDE is drastically reduced, and applications of standard minimization algorithms for the calibration are significantly faster than working with a high-dimensional finite element basis. Secondly, so-called de-Americanization strategies are applied. Here, the main idea is to reformulate the calibration problem for American options as a problem for European options and to exploit closed-form solutions. Both reduction techniques are systematically compared and tested for both synthetic and market data sets.

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Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation

Cited by 9 publications

References 47 publications

Reduced-Order Modeling for Heston Stochastic Volatility Model

Reduced-Order Modeling for Heston Stochastic Volatility Model

Modulus-based Successive Overrelaxation Iteration Method for Pricing American Options with the Two-asset Black–Scholes and Heston's Models Based on Finite Volume Discretization

Complexity reduction for calibration to American options

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